what are the necessary conditions for a weighted directed hypergraph to have an incidence matrix of full rank?
In this context, we can define the incidence matrix as follows:
Let $V = \{v_1,v_2,...,v_m\}$ be the set of vertices and $E = \{e_1,e_2,...,e_n\}$ be the set of hyperedges (possibly connecting more than two vertices) of the weighted directed hypergraph $H$. The $m \times n$ incidence matrix is $A = (a_{ij})$ where
$$ a_{ij} = \begin{cases} w_{ij}, & \text{if}\ v_{i} \in e_j \\ 0, & \text{otherwise} \end{cases} $$
where $w_{ij} \ne 0$ is the weight that can be negative because of the direction of the hyperedge. Each column of $A$ has at least one positive entry and one negative entry, so it has at least two entries.
For directed graphs it is well known that the rank of the incidence matrix is equal to $m - c$, where $c$ is the number of connected components of the graph. Lets consider that the hypergraph $H$ has only one connected component ($c = 1$).
This question is a generalization of this one: Which graphs have incidence matrices of full rank?
